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Calabi quasimorphisms for the symplectic ball

Paul Biran·Michael Entov·Leonid Polterovich
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Abstract

We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on any symplectomorphism supported in any "sufficiently small" open subset of the ball equals the Calabi invariant of the symplectomorphism. By a "sufficiently small" open subset we mean that it can be displaced from itself by a symplectomorphism of the ball. As a byproduct we show that the (Lagrangian) Clifford torus in the complex projective space cannot be displaced from itself by a Hamiltonian isotopy.

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Paul Biran, Michael Entov, Leonid Polterovich. 2003-10-29. Calabi quasimorphisms for the symplectic ball. https://arxiv.org/abs/math/0307011

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